Using First-Principles Simulations to Discover Materials with Ultra-low Work Functions for Energy Conversion Applications Sharon
1,2 Chou* ,
Igor
1,2 Bargatin ,
Frank
3 Abild-Pedersen ,
Motivation
Aleksandra
3 Vojvodic ,
Piero
2,4 Pianetta ,
Work function, surface dipole, surface coverage
We are interested in the discovery of new materials with low work functions because the efficiency of both traditional thermionic energy converters (TECs) and photon-enhanced thermionic emission (PETE) converters depends critically on the work function of the anode (φc in figure). For anodes rejecting heat near room temperature, the optimal anode work function is approximately Tanode/(700 K) = 0.5 eV [Hatsopoulos1973].
Charge transfer from adsorbates affects work function (Langmuir-Gurney model)
Work function is linearly related to surface dipole Surface dipole that increases work function
J E AT 2e E / kT
Surface dipole that lowers work function
Roger T.
1,2 Howe
Tunneling probabilities and net charge densities 1 Cs atom adsorbed per (3x4) W(bcc110) unit cell, Coverage θ ≈ 38%, φ ≈ 2.10 eV Electron tunneling probabilities 10%
Work function decrease
At low surface coverage
Vout E C
Materials with such low work functions have not been discovered yet. Therefore, thermionic converters typically use the anodes with the lowest work functions available, such as cesiated tungsten anodes with work functions ≈ 1.5 eV. This means that traditional thermionic converters could have appreciable efficiencies only at heat source temperatures above 1500 °C.
Jens K.
3 Nørskov ,
Net charge density isosurfaces
Isosurfaces contour spacing ~1e-3 q/Å3
1%
Work function minimum
0.1%
At medium surface coverage
Blue – net positive (electron poor) maximum ~0.02 q/Å3 Red – net negative (electron rich) maximum ~0.03 q/Å3
1 Ångstrom
A new mechanism - photon-enhanced thermionic emission (PETE) - can be particularly useful for solar energy applications as it can use both the high per-photon energy of solar photons and the heat resulting from sub-bandgap photons and other losses. Similar to conventional TECs, the efficiency of PETE converters is strongly dependent on the work function of the anode. For anode work function of ~0.5 eV, the efficiency of PETE converters can exceed 60% [Schwede2010].
5Å
Work function rebound dipole areal density change
work function change
(unit = electron charge·Å)
Effect of various adsorbates on dipole areal density and work function
Work function exhibits minimum at partial surface coverage (e.g. Cs on W) DFT simulation •
At low coverage, dipole decreases Ф on metal substrate
•
Depolarization at higher coverage than minimum, Ф rebounds
Li Na
Na
Li
Taylor & Langmuir (1930’s) experimental data
Cs Na
5Å
Isosurfaces contour spacing ~1.5e-3 q/Å3
0.1%
Li
Blue – net positive (electron poor) maximum ~0.05 q/Å3 Red – net negative (electron rich) maximum ~0.04 q/Å3
1 Ångstrom
Cs & top layer of W relaxed, Cs ~3.9Å above W surface, valence charge density difference profiles & planar averages, units = q/Å3
Cs
Left: Levelized cost of electricity (LCOE) for a solar thermal power plant with a thermionic topping cycle versus the incremental cost of thermionic/PETE topping stage for three values of its efficiency. The 2005 LCOE for conventional power plants [BV2007] are shown for comparison. The dashed circle highlights that solar thermal electricity becomes cheaper than coal with a 35%-efficient thermionic topping stage that costs less than $1/Watt.
10%
Net charge density isosurfaces
1%
W(110) bare surface Φ ≈ 4.68 eV
ΔΦ (eV)
Electron tunneling probabilities
O
O
W(100) bare surface Φ ≈ 3.90 eV
2 Cs atoms adsorbed per (3x4) W(bcc110) unit cell, Coverage θ ≈ 75%, φ ≈ 1.41 eV
At high surface coverage
Cs fixed at 2.6Å above W surface, valence charge density difference profiles & planar averages, units = q/Å3
Cs
W(110)
W(110)
-Δp (q/Å)
Right: Calculated efficiency limit for a thermionic energy converter as a function of the cathode temperature for three values of the collector (anode) work function [Hatsopoulos1973]. The dashed curves show the Carnot efficiency limit, limits for a converter with a figure of merit ZT = 2 (roughly corresponds to the best existing thermoelectric materials), and ZT = 10 (much better than the current state of the art). The heat sink is assumed to be at room temperature (300 K) in all cases.
_
Electrostatic potential profiles Electrostatic potential profile averaged on each X-Y plane, eCs on W(110)
Electrostatic potential (eV)
Density functional theory and implementation Density functional theory (DFT) solves Schrödinger’s equation to approximately model the ground-state energetics of atomic systems
N = number of electrons Kinetic energy of electrons
Electron-nuclei interactions
Electron-electron interactions
+
+ _
• Potential barrier caused by Cs atom on top of 4 W layers
_
• Barrier height and width affect electron tunneling probability → emissivity → electric current conduction • Emissivity is related to the Richardson-Dushman constant, denoted A Current density • Work function φ = Vacuum level potential (Vvac) – Fermi level (Ef)
ri = spatial vector of the ith electron Supercell Z height (Ångstrom), larger height → farther from substrate surface
Electrostatic potential profile at Y = 1.141Å, Cs on W(100)
Trajectory of minimum potential points on each X-Y plane, Cs on W(100)
Similar contours as the W(110) case, where W layers are in a lowpotential zone and Cs atoms are in a highpotential zone.
The trails of low potential “wrap” around the Cs atoms.
Net dipole points up (negative charge closer to the W surface) even though there is a smaller “counter-dipole” pointing down. Work function Φ ≈ 1.7eV.
No counter-dipole in this case where Cs atom is closer to the W surface. This configuration generates a larger up-pointing dipole and a lower work function (~1.3eV), but is not energetically favorable.
• Ground-state energy E[n(r)] is a functional of electron density • Can calculate work function from n(r)
[HK1964]
DFT calculations for solid surfaces: Slab model with periodic boundary conditions in 2D • Tiling supercells → slab • Supercell size determined by the minimum volume that captures necessary atomic properties • Needs enough vacuum space to keep electrons’ wave functions from each slab isolated
Supercell of Cs atoms on tungsten (W) Atoms free to move Atoms fixed by simulator
Four W layers extend infinitely in 2D
{
Layer 1 Layer 2 Layer 3 Layer 4
{ 1
Emitted electrons are likely to pass through low-potential regions.
These are paths that emitted electrons are likely to follow.
Authors’ affiliations and acknowledgement *
[email protected] 1Center of Interfacial Engineering for Microelectromechanical Systems (CIEMS) at Stanford 2Department of Electrical Engineering 3SUstainable eNergy through CATalysis (SUNCAT) center 4Stanford Synchrotron Radiation Lightsource (SSRL)
References [BV2007] Black and Veatch corporation, “20 percent wind energy penetration in the United States”, Final Report Project Number 144864 [Hatsopoulos1973] G. N. Hatsopoulos and E. P. Gyftopoulos, “Thermionic Energy Conversion”, vols. 1 (Cambridge, MA.: MIT Press, 1973) [HK1964] Hohenberg P, Kohn W. Inhomogeneous electron gas. Physical Review 136 (3B): B864–B871. (1964) [Schwede2010] J.W. Schwede, I. Bargatin, D.C. Riley, B.E. Hardin, S.J. Rosenthal, Y. Sun, F. Schmitt, P. Pianetta, R.T. Howe, Z.-X. Shen, N.A. Melosh, Nature Mat., 9 762 (2010)
